A076440 Numbers k which appear to have a unique representation as the difference of two perfect powers where one of those powers is odd; that is, there is only one solution to Pillai's equation a^x - b^y = k, with a > 0, b > 0, x > 1, y > 1 and that solution has odd x or odd y (or both odd).

1, 2, 10, 30, 38, 46, 122, 126, 138, 142, 146, 150, 154, 166, 170, 190, 194, 214, 222, 234, 270, 282, 298, 318, 338, 342, 354, 370, 382, 386, 406, 486, 490, 498, 502, 518, 546, 550, 566, 574, 582, 586, 594, 638, 666, 678, 686, 694, 710, 726, 730, 734, 746

Offset

1,2

Comments

There are two types of unique solutions. See A076438 for the general case. This sequence was found by examining all perfect powers (A001597) less than 2^63-1. By examining a larger set of perfect powers, we may discover that some of these numbers do not have a unique representation.

References

R. K. Guy, Unsolved Problems in Number Theory, D9.

T. N. Shorey and R. Tijdeman, Exponential Diophantine Equations, Cambridge University Press, 1986.

Links

Table of n, a(n) for n=1..53.

M. E. Bennett, On Some Exponential Equations Of S. S. Pillai, Canad. J. Math. 53 (2001), 897-922.

T. D. Noe, Unique solutions to Pillai's Equation requiring an odd power for n <= 1000.

Eric Weisstein's World of Mathematics, Pillai's Conjecture.

Crossrefs

Cf. A001597, A076438, A076439.

Sequence in context: A098425 A098408 A063564 * A047198 A290461 A162524

Adjacent sequences: A076437 A076438 A076439 * A076441 A076442 A076443

Keyword

hard,nonn

Author

T. D. Noe, Oct 12 2002

Status

approved